(square roots of the identity matrix) For how many 2×2 matrices A is it true that A2=I ? Now answer the same question for n×x n matrices where n&gt;2

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(square roots of the identity matrix) For how many 2×2 matrices A is it true that A2=I ? Now answer the same question for n×x n matrices where n&amp;gt;2

au4gsf · Accepted Answer

Step 1 To find how many matricec A of order 2 exists such that A2=I Here we have to find A2=A×A=I Let us consider any arbitrary matrix of order 2. A=(abcd) Then A2 is given by A2=(abcd)(abcd)=(a2+bcab+bdca+dcd2+bc) Now from A2=I , we get A2=(a2+bcb(a+d)c(a+d)d2+bc)=(1001)=I (1)a2+bc=1 (2)d2+bc=1 (3)b(a+d)=0 (3)c(a+d)=0 Step 2 Case 1: If (a+d)≠0 From (3) we get : b=0 From (4) we get : c=0 From (1) we get : a2=1⇒a=±1 From (2) we get : d2=1⇒d=±1 Therefore we get the matrix of the form {A=(a00d):a=±1,d=±1} Therefore there are only 4 matrices exist such that A2=I from case 1. Step 3 Case 2: If (a+d)=0 sub case a: If a=0 , that implies d=0 Then from(1 ) and (2) , we get : bc=1⇒c=b−1=1b in real numbers. Therefore we get the matrix of the form {A=(0aa−10):a∈F} Therefore there are infinitely many matrices exist if the field is the set of real numbers such that A2=I Step 4 We can also consider further subcases to find the type of involuntary matrices, but as the question is asked to find how many, we have already got infinitely many 2×2 matrices of such type. Similarly for any n&amp;gt;2 , we can find infinitely many matrices satisfying this condition on the field of real numbers.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

(square roots of the identity matrix) For how many 2x2 matrices A is it true that A^2=I ? Now answer the same question for n x n matrices where n>2

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